How Do I Bring Up The Same Properties Menu As In This Video?
The author of the video is adjusting the parameters of a newly added object.

In 3d view, add a new object and press F6. But there is a catch: if you do anything else other than press F6 after creating it, you don't get the window.



Many tutorials I've seen over the years teach you to do this through tool shelf, which isn't visible in that video.